| L(s) = 1 | − i·3-s + (1.48 + 1.67i)5-s + i·7-s − 9-s − 6.31·11-s − 6.96i·13-s + (1.67 − 1.48i)15-s + 6.57i·17-s − 3.73·19-s + 21-s + 5.73i·23-s + (−0.612 + 4.96i)25-s + i·27-s + 2·29-s + 1.03·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (0.662 + 0.749i)5-s + 0.377i·7-s − 0.333·9-s − 1.90·11-s − 1.93i·13-s + (0.432 − 0.382i)15-s + 1.59i·17-s − 0.857·19-s + 0.218·21-s + 1.19i·23-s + (−0.122 + 0.992i)25-s + 0.192i·27-s + 0.371·29-s + 0.186·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6255701579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6255701579\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-1.48 - 1.67i)T \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 + 6.31T + 11T^{2} \) |
| 13 | \( 1 + 6.96iT - 13T^{2} \) |
| 17 | \( 1 - 6.57iT - 17T^{2} \) |
| 19 | \( 1 + 3.73T + 19T^{2} \) |
| 23 | \( 1 - 5.73iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 1.03T + 31T^{2} \) |
| 37 | \( 1 - 10.7iT - 37T^{2} \) |
| 41 | \( 1 + 6.96T + 41T^{2} \) |
| 43 | \( 1 + 5.92iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 1.03iT - 53T^{2} \) |
| 59 | \( 1 + 3.22T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 4.77iT - 67T^{2} \) |
| 71 | \( 1 + 8.23T + 71T^{2} \) |
| 73 | \( 1 + 4.26iT - 73T^{2} \) |
| 79 | \( 1 + 5.92T + 79T^{2} \) |
| 83 | \( 1 - 3.22iT - 83T^{2} \) |
| 89 | \( 1 + 2.18T + 89T^{2} \) |
| 97 | \( 1 - 3.73iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.938354255234116218907455598776, −8.538604960543851814293977950064, −8.056093589845106043913664984747, −7.38392543707331839432484556545, −6.25230382264972743836373607240, −5.72757461384280251470965652158, −5.04062691819215566949247818014, −3.30987063563116237682606979886, −2.72872323708916814607659445729, −1.68771914612339091273721285716,
0.21429298850769099316222447182, 2.00503786719265638891576340206, 2.84820395713762471101419538693, 4.53228677761958922868746905640, 4.61897462494894681333250983222, 5.66330085344481926366313201118, 6.60905229423063489631935744433, 7.50597449354321976084908849805, 8.499293400610553827840266446100, 9.109336675129902428612616631377
